where \(\Pi _{\mathcal {N},ode}^{S}\)  denotes the projection operator used to define the reduced problem for the ODE (REF ), see [1]}. It follows by Proposition REF  that the reduced map is
\(z \mapsto \bar{z} = z + \varepsilon h \Pi _{\mathcal {N}}^{S} G(z,0) |_S = z + \varepsilon h \Pi _{\mathcal {N},ode}^{S} G(z,0)|_S ,\) 
